A study on the Hamilton-Jacobi equation arising from nonlinear control theory

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Firstly, the theory of partialdifferential equations of the 1st order is outlined in the context of the Hamilt on-Jacobi equation. According to the theory, to solve the Hamilton-Jacobi equation, all one has to do is solve a system of ordinary differential equations and/orintegrate a completely integrable Pfaffian system.Secondly, attention is paid to a special kind of solution, a stabilizing solution, which plays an important role in control theory. The well-known theory by Arimoto and Potter, that is, a necessary and sufficient condition of the existence of a stabilizing solution, is reviewed as a part of the theory of the Hamilton-Jacobi equation. In control theory, we are interested in obtaining as many solutions as possible. To do this, more information on the structure of the Hamilton-Jacobi equation will be necessary, which is indicated by an example.Finally, the theory of the generating function for symplectic transforms is introduced ; this will play a central role in a later subsection. Given a solution to the Hamilton-Jacobi equation, an auxiliary equation is derived that determines the other solutions using the generating function of symplectic transforms. It will also be seen that the auxiliary equation even characterizes the whole structure of the Hamilton-Jacobi equation. The linear control theoretic explanation of this structure will also be provided.